Effectiveness factor catalyst

The catalyst effectiveness factor can be estimated as a function of the Thiele modulus (O). The generalized Thiele modulus for nth-order irreversible reaction is (Froment et al., 2010) 

For nth-order reversible reaction, the generalized Thiele modulus is 

The equivalent particle diameter was estimated using the following expression proposed by Cooper et al. (1986)  

The tortuosity factor (x) generally has a value of 2-7. Usually, tortuosity factor is assumed to be 4 according to the literature reports (Bird et al., 2002). According to Satterfield (1970), Knudsen diffusion is not observed in liquids therefore, = 0. 

The following equations are employed for determining the values of tj for each reaction (Dudukovid, 1977 Tarhan, 1983 Froment et al., 2010)  

---

This section is concerned with analyses of simultaneous reaction and mass transfer within porous catalysts under isothermal conditions. Several factors that influence the final equation for the catalyst effectiveness factor are discussed in the various subsections. The factors considered include different mathematical models of the catalyst pore structure, the gross catalyst geometry (i.e., its apparent shape), and the rate expression for the surface reaction. 

The equations for effectiveness factors that we have developed in this subsection are strictly applicable only to reactions that are first-order in the fluid phase concentration of a reactant whose stoichiometric coefficient is unity. They further require that no change in the number of moles take place on reaction and that the pellet be isothermal. The following illustration indicates how this idealized cylindrical pore model is used to obtain catalyst effectiveness factors. 

ILLUSTRATION 12.2 DETERMINATION OF CATALYST EFFECTIVENESS FACTOR FOR THE CUMENE CRACKING REACTION FROM MEASUREMENT OF AN APPARENT RATE CONSTANT... 

The Effectiveness Factor Analysis in Terms of Effective Diffusivities First-Order Reactions on Spherical Pellets. Useful expressions for catalyst effectiveness factors may also be developed in terms of the concept of effective diffusivities. This approach permits one to write an expression for the mass transfer within the pellet in terms of a form of Fick s first law based on the superficial cross-sectional area of a porous medium. We thereby circumvent the necessity of developing a detailed mathematical model of the pore geometry and size distribution. This subsection is devoted to an analysis of simultaneous mass transfer and chemical reaction in porous catalyst pellets in terms of the effective diffusivity. In order to use the analysis with confidence, the effective diffusivity should be determined experimentally, since it is difficult to obtain accurate estimates of this parameter on an a priori basis. 

Illustration 12.3 indicates the use of the effective diffusivity approach for estimating catalyst effectiveness factors when this parameter is determined experimentally or may be estimated. 

Effectiveness Factors for Reversible Reactions. The vast majority of the literature dealing with catalyst effectiveness factors pre-... 

The Consequences of Intraparticle Temperature Gradients For Catalyst Effectiveness Factors... 

The following illustration indicates how experimental and calculated values of catalyst effectiveness factors may be determined for a specific case. 

This relation is plotted as curve Bin Figure 12.11. Smith (66) has shown that the same limiting forms for are observed using the concept of effective dififusivities and spherical catalyst pellets. Curve B indicates that, for fast reactions on catalyst surfaces where the poisoned sites are uniformly distributed over the pore surface, the apparent activity of the catalyst declines much less rapidly than for the case where catalyst effectiveness factors approach unity. Under these circumstances, the catalyst effectiveness factors are considerably less than unity, and the effects of the portion of the poison adsorbed near the closed end of the pore are not as apparent as in the earlier case for small hr. With poisoning, the Thiele modulus hp decreases, and the reaction merely penetrates deeper into the pore. 

If the two competing reactions have the same concentration dependence, then the catalyst pore structure does not influence the selectivity because at each point within the pore structure the two reactions will proceed at the same relative rate, independent of the reactant concentration. However, if the two competing reactions differ in the concentration dependence of their rate expressions, the pore structure may have a significant effect on the product distribution. For example, if V is formed by a first-order reaction and IF by a second-order reaction, the observed yield of V will increase as the catalyst effectiveness factor decreases. At low effectiveness factors there will be a significant gradient in the reactant concentration as one moves radially inward. The lower reactant concentration within the pore structure would then... 

When the hydrogen pressure is 1 atm, and the temperature is 77 °K, the experimentally observed (apparent) rate constant is 0.159 cm3/ sec-g catalyst. Determine the mean pore radius, the effective diffusivity of hydrogen, and the catalyst effectiveness factor. 

At this point it is instructive to consider the possible presence of intraparticle and external mass and heat transfer limitations using the methods developed in Chapter 12. In order to evaluate the catalyst effectiveness factor we first need to know the combined diffusivity for use... 

From Figure 12.2 it is evident that the catalyst effectiveness factor for isothermal operation will be approximately 0.47. At higher temperatures the effectiveness factor will be smaller because the rate constant will increase more rapidly with temperature than will the combined diffusivity. However, the reactions in question are quite... 

Figure 13.6 indicates that very large temperature gradients exist near the beginning of the bed and that the higher the inlet temperature, the greater the difference between this temperature and the maximum temperature achieved in the bed. Catalyst effectiveness factor profiles mirror the temperature profiles in an opposite... 

Z distance from tubular reactor inlet rj catalyst effectiveness factor... 

Derive an expression for the catalyst effectiveness factor (17) for a spherical catalyst particle of... 

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

The study of the intra-phase mass transfer in SCR reactors has been addressed by combining the equations for the external field with the differential equations for diffusion and reaction of NO and N H 3 in the intra-porous region and by adopting the Wakao-Smith random pore model to describe the diffusion of NO and NH3 inside the pores [30, 44]. The solution of the model equations confirmed that steep reactant concentration gradients are present near the external catalyst surface under typical industrial conditions so that the internal catalyst effectiveness factor is low [27]. 

What equation would have to be solved for the catalyst effectiveness factor for a second-order reaction What is the apparent aetivation energy of a second-order reaction in the limit of... 

The importance of the wetting efficiency results mainly from the fact that it is closely related to the reaction yield, and more specifically to the catalyst effectiveness factor (Burghardt et al., 1995). The reaction rate over incompletely covered catalytic particles can be smaller or greater than the rate observed on completely wetted packing, depending on whether the limiting reactant is present only in the liquid-phase or in both gas and liquid-phases. 

To sum up, special attention should be given to the effect of temperature on the process during the design and control of commercial catalytic reactors. Moreover, the same size of catalyst particles should be used at any scale so that the catalyst effectiveness factor also remains the same. The available catalytic surface per reactor volume and the space velocity (when the rate is not controlled by mass transfer phenomena) should also be left unchanged at any scale. 

Since the cooling jacket has cocurrent flow, the model consists of the set of four coupled initial value differential equations (7.5) to (7.8). Note that the first three DEs (7.5) to (7.7) contain the variable catalyst effectiveness factor rj. Thus there are other equations to be solved at each point along the length 0 < / < Lt of the reactor tube, namely the equations for the catalyst pellet s effectiveness factor rj. 

The degree of conversion has to be considerably higher than 50% to obtain the desired product in sufficiently high enantiomeric purity. The incomplete enantio-selectivity of PPL (E = 18) was countered by continuation of the reaction to 67% conversion. The productivity in a multi-phase reactor (Figure 13.24), in which the racemic ester was circulated in the lumen, was 17.6 g (h m2)-1 or 28.4 mmol (h mg enzyme)-1. The main problem of the multi-phase reactor is the low catalyst effectiveness factor, which is normally found to lie between 30 and 50%. 

Figure 4a and 4b. Catalyst effectiveness factor as a function of thiele modulus... 

The quotient from the effective reaction rate (reff) and that (r) without diffusion control (cG = cs) defines the outer catalyst effectiveness factor, r ( Xt ... 

Fig. 80 External catalyst effectiveness factor tie as a function of the Damkohler-ll number DaM s k/ kGa and the reaction order m from [118].

---